$k$-quasi $n$-power posinormal Weighted Composition and Cauchy Dual of Moore-Penrose inverse of Lambert Operators

In this paper we characterize \(k\)-quasi \(n\)-power posinormal composition operators and weighted composition operators on the Hilbert space \(L^2(Σ)\). For Lambert conditional operators (of the form \(T = M_w E M_u\)), we establish necessary and sufficient conditions under which these Cauchy duals via the Moore-Penrose inverse become \(k\)-quasi \(n\)-power posinormal operators. Finally, we construct an explicit example of a \(k\)-quasi \(n\)-power posinormal weighted shift operator on a rooted directed tree.